Linear Function Facts That Will Make You Feel Much Smarter
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One of the most interesting aspects of mathematics is a linear function.
Represented by a straight line on a graph, a linear function is quite useful. It is written as: y= f(x) = a + bx.
Linear functions and linear equations are often confused, although they are two different concepts. To fully understand the concept of a linear function, it is essential to differentiate between the two.
Interestingly, linear functions have many practical uses. From calculating the rate of change to calculating finances, linear functions can be used to solve it all.
Knowing the right concepts and equations can help you pinpoint a linear function quite easily. There are similarities between finding the equation of a line and finding a linear function.
To know more interesting linear function facts, continue reading!
Types Of Linear Functions
To begin with, linear functions are nothing but a combination of a dependant and an independent variable whose graph is a straight line. Linear functions cannot be sub-divided into categories, but linear equations can be broadly categorized into four types. These are direct variation, standard form, slope-intercept form, and point-slope form.
Direct variations contain a pair of variables that are directly proportional to each other. This is usually represented in the general slope-intercept form.
A standard form of a linear equation contains two independent variables, namely x and y, and is represented by ax + by = c. Here a, b, and c are positive and negative integers having finite values.
The slope-intercept form is usually defined as the equation used in calculating the slope of a straight line by taking into consideration its intercepts, or when the line crosses the x and y-axis in graphs.
The point-slope form is used to represent the straight-line ax + by = c using its two points x and y as well as its slope m.
Linear Function Vs. Linear Equation
Linear functions and linear equations are interdependent concepts that are useful in representing lines in a graph. However, some fundamental differences should be considered while defining the two points.
In a linear function, values are inserted to get the output. In a linear equation, the input value is obtained for a predetermined output.
A linear function can be written as a combination of more than one variable, and each of the terms are essentially a polynomial of degree one. It has one or two variables without exponents. So, f(x) = 7x + 9y is a classic example of a linear function.
Each of the terms is either a constant or the product of a constant of one variable. So, to conclude, linear equations can be used to describe linear functions.
In a linear equation, however, the highest power of the variable is one in simple algebraic expressions. It is also defined as an equation of degree one. The graph obtained from a linear equation is always a straight line. Most linear equations constitute a range of functions. For every value of x, there is a corresponding value of y.
Practical Usage Of Linear Functions
Like most other aspects of mathematics, a linear function can be used in your day-to-day life. In the real world, a linear function can be applied to situations ranging from change of rates (at a constant rate) to pricing problems.
These days, most people use movie streaming services. A linear function can be applied in this scenario while calculating the total fee you have to pay your service provider.
For instance, if the streaming service provider has a fixed monthly charge of $5.00, and charges $0.50 for every movie that has been downloaded, then the total fee that needs to be paid can be represented as f(x) = 0.5x + 5, where x stands for the number of movies that have been downloaded that month.
With a linear function it is also possible to incorporate multiple equations.
This usage of a linear function can be easily demonstrated with a problem about moving trains.
If trains A and B are 500 mi (804.6 km) away from each other, and if train A is moving towards train B with a speed of 50 mph (80.4 kph), while the latter is moving at a speed of 80 mph (128.7 kph) towards train A, then with the help of a linear function, you can easily determine at what time the trains will meet and also measure the distance they traveled to meet at that point in time.
All you have to do is input values in the equation: y = mx + b.
Subsequently, graphs can also be used to draw the two lines, and the point where the lines cross each other will give you the required distance and time values.
FAQs
What are the characteristics of a linear function?
The main characteristic of a linear function is that it represents a straight line. Furthermore, it has two variables, namely, an independent variable and a dependent variable. The independent variable is x and the dependent variable is y.
What are the three key features of a linear function?
The characteristic form of a linear function is written as: y= f(x)= a+bx. Here, x represents the independent variable, whereas y is the dependent variable. Furthermore, b stands for the coefficient of the independent variable.
How do you determine a linear function?
A linear function can be determined by comparing it with the equation y = mx+b. This is because linear functions are in this form, while non-linear functions are not.
What does a linear function represent?
A linear function is used to represent a straight line.
How do you find the domain of a linear function?
The method to determine the domain of a linear function is pretty straightforward. To do so, all you have to do is identify the x-coordinate sets present on the function's graph.
What does a linear function look like?
On a coordinate plane, a linear function would resemble a straight line.
How do you find a linear function?
It is a simple process to find a linear function. Usually, the slope-intercept form and the point-slope form are used to pinpoint a linear function. Overall, the process is quite similar to how the equation of a line can be found.
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